论文标题
Connes的双中心化问题，用于Q构造的Araki-Woods代数
Connes' bicentralizer problem for q-deformed Araki-Woods algebras
论文作者
论文摘要
令$（H _ {\ Mathbf {r}}，U_T）$为$ \ Mathbf {r} $在真实（可分开的）Hilbert Space $ H _ {\ Mathbf {R MathBf {R}} $上的任何强烈连续的正交表示。对于任何$ q \ in（-1,1）$，我们用$γ_Q（h_ {\ m马理{r}}，u_t）^{\ prime \ prime} $ $ q $ q $ q $ q $ q $ q $ q $ q $ q $ q $ q $ q $ q-deformed araki-woods algebra由Shlyakhtenko and Hiai引入。在本文中，我们证明$γ_Q（h _ {\ Mathbf {r}}，u_t）^{\ prime \ prime} $如果是类型$ \ rm iii_1 $ factor，则具有琐碎的双肠词。特别是，我们获得了$γ_Q（h _ {\ mathbf {r}}，u_t）^{\ prime \ prime} $始终承认最大的Abelian subelgebra，这是忠实的正常条件期望的范围。此外，使用Sniesdy的工作，我们得出$γ_Q（H _ {\ MathBf {r}}，U_T）^{\ Prime \ Prime} $是一个完整的因素，前提是$（H _ {\ MathBf {r}}，r}}，u_t）$的弱混合部分是nyzero的。
Let $(H_{\mathbf{R}}, U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ has trivial bicentralizer if it is a type $\rm III_1$ factor. In particular, we obtain that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady's work, we derive that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ is a full factor provided that the weakly mixing part of $(H_{\mathbf{R}}, U_t)$ is nonzero.
